Theorem xpdisj1 5028

Description: Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Assertion

Ref	Expression
xpdisj1	|- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Proof of Theorem xpdisj1

Step	Hyp	Ref	Expression
1		inxp 4738	. 2 |- ( ( A X. C ) i^i ( B X. D ) ) = ( ( A i^i B ) X. ( C i^i D ) )
2		xpeq1 4618	. . 3 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = ( (/) X. ( C i^i D ) ) )
3		0xp 4684	. . 3 |- ( (/) X. ( C i^i D ) ) = (/)
4	2, 3	eqtrdi 2215	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = (/) )
5	1, 4	syl5eq 2211	1 |- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1343   i^i cin 3115   (/)c0 3409   X. cxp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by:  djudisj  5031  xp01disjl  6402  xpfi  6895  djuinr  7028